Third-Order Nonlinear Semi-Canonical Functional Differential Equations: Oscillation via New Canonical Transform

Abstract

This paper focuses on the oscillatory properties of the third-order semi-canonical nonlinear delay differential equation. By using the new canonical transform method, we transformed the studied equation into a canonical-type equation, which simplified the examination of the studied equation. The obtained oscillation results are new and complement the existing results mentioned in the literature. Examples are provided to illustrate the importance and novelty of the main results.

1. Introduction

The aim of this paper is to present new oscillation criteria for the third-order delay differential equation:

$$\mathfrak{L}\zeta \left(t\right)+\eta \left(t\right){\zeta }^{\alpha }\left(\tau \left(t\right)\right)=0,t\ge t_0>0,$$

(E)

where $\mathfrak{L}$ is the differential operator defined by

$$\mathfrak{L}\zeta \left(t\right)={\left({\delta }_2\left(t\right){\left({\delta }_1\left(t\right){\zeta }^{\prime }\left(t\right)\right)}^{\prime }\right)}^{\prime }.$$

(1)

In the sequel, we assume the following hypotheses:

$\left(H_1\right)$

${\delta }_1\in C^2\left(\right[t_0,\infty ),R),$ ${\delta }_2\in C^1\left(\right[t_0,\infty ),R),$ ${\delta }_1\left(t\right)>0$ and ${\delta }_2\left(t\right)>0$ for all $t\ge t_0;$

$\left(H_2\right)$

$\eta \left(t\right)\in C\left(\right[t_0,\infty ),R)$ and $\eta \left(t\right)>0$ for all $t\ge t_0;$

$\left(H_3\right)$

$\tau \left(t\right)\in C^1\left(\right[t_0,\infty ),R),$$\tau \left(t\right)<t,$${lim}_{t\to \infty }\tau \left(t\right)=\infty$ and ${\tau }^{\prime }\left(t\right)>0$ for all $t\ge t_0;$

$\left(H_4\right)$

$\alpha$ is a ratio of odd positive integers;

$\left(H_5\right)$

the operator $\mathfrak{L}$ is in a semi-canonical form, that is,

$${\int }_{t_0}^{\infty }{\displaystyle \frac{1}{{\delta }_2\left(t\right)}}dt=\infty and{\mathrm{\Omega }}_1\left(t_0\right)={\int }_{t_0}^{\infty }{\displaystyle \frac{1}{{\delta }_1\left(t\right)}}dt<\infty .$$

Definition 1.

The solution of (E) is defined to be a nontrivial function $\zeta \left(t\right)\in C\left([t_{\zeta },\infty )\right),$$t_{\zeta }\ge t_0$ with the properties ${\delta }_1{\zeta }^{\prime }\in C^1\left([t_{\zeta },\infty )\right),$${\delta }_2{\left({\delta }_1{\zeta }^{\prime }\right)}^{\prime }\in C^1\left([t_{\zeta },\infty )\right),$ which satisfies (E) on $[t_{\zeta },\infty ).$ We consider only the solutions $\zeta \left(t\right)$ of (E) satisfying $sup\left\{\right|\zeta \left(t\right)|:t\ge T\}>0$ for all $T\ge t_{\zeta }$ and assume that (E) possesses such a solution.

Definition 2.

A solution of (E) is called oscillatory if it has infinitely many zeros on $[t_{\zeta },\infty );$ otherwise, it is called nonoscillatory. Equation (E) itself is called oscillatory if all its solutions are oscillatory.

Due to many practical important applications of third-order functional differential equations, as well as a number of mathematical problems involved [1], the area of the qualitative theory of such equations received a great attention in the recent several decades. Further, it is very useful in predicting the similar behavior of solutions of third-order partial differential equations [2,3]. For instance, in many interesting physical phenomena, the Kuramoto–Sivashinsky equation

$$z_t+z_{xxxx}+z_{xx}+{\displaystyle \frac{1}{2}}{z}^2=0$$

plays a certain role; it is used to describe pattern formulations in reaction diffusion systems and to model the instability of flame front propagation (see [4,5]). To find the traveling wave solutions of this partial differential equation, one may use the substitution of the form $z(x,ct)=z(x-ct)$ with speed c and solve a third-order nonlinear differential equation

$$\lambda z^{‴}\left(x\right)+z^{\prime }\left(x\right)+f\left(z\right)=0.$$

Hence, it is interesting to investigate the qualitative theory of a third-order differential equation of the form

$$y^{‴}+p\left(t\right)y^{\prime }\left(t\right)+q\left(t\right)y^{\alpha }\left(\sigma \left(t\right)\right)=0.$$

Note that the above equation can be included as a particular case of Equation (E).

Oscillatory phenomena play a significant role in understanding the inherent vibrational patterns within dynamic systems, so investigating the oscillatory behavior of solutions to differential equations provides valuable insights into the stability and periodicity of the systems under consideration. In recent years, there has been great interest in investigating the oscillatory behavior of solutions of (E) and their particular cases or generalizations (see, for example, the monographs [2,3,6], as well as the papers [7,8,9,10,11,12,13,14,15,16] and the references cited therein).

In the papers [9,10,12,13,14,15,16], the authors established sufficient conditions for the oscillation behavior of solutions of Equation (E) and their particular cases or generalizations under the following conditions:

$${\int }_{t_0}^{\infty }{\displaystyle \frac{1}{{\delta }_2\left(t\right)}}dt={\int }_{t_0}^{\infty }{\displaystyle \frac{1}{{\delta }_1\left(t\right)}}dt=\infty ,$$

or

$${\int }_{t_0}^{\infty }{\displaystyle \frac{1}{{\delta }_2\left(t\right)}}dt<\infty and{\int }_{t_0}^{\infty }{\displaystyle \frac{1}{{\delta }_1\left(t\right)}}dt=\infty ,$$

or

$${\int }_{t_0}^{\infty }{\displaystyle \frac{1}{{\delta }_2\left(t\right)}}dt<\infty and{\int }_{t_0}^{\infty }{\displaystyle \frac{1}{{\delta }_1\left(t\right)}}dt<\infty .$$

On the other hand, in [11,17,18], the authors considered Equation (E) or its generalizations under the condition

$${\int }_{t_0}^{\infty }{\displaystyle \frac{1}{{\delta }_2\left(t\right)}}dt=\infty and{\int }_{t_0}^{\infty }{\displaystyle \frac{1}{{\delta }_1\left(t\right)}}dt<\infty$$

and established several criteria which imply that the solutions are either oscillatory or converge to zero. However, in [17,18], the authors obtained their oscillation results by transforming Equation (E) into a canonical form under the assumption that

$${\mathrm{\Omega }}_{21}\left(t_0\right)={\int }_{t_0}^{\infty }{\displaystyle \frac{{\mathrm{\Omega }}_1\left(t\right)}{a_2\left(t\right)}}dt=\infty .$$

(2)

This greatly simplified the examination of (E) by reducing the set of nonoscillatory solutions into two instead of three. So, the authors obtained the oscillation criteria by eliminating these two types of nonoscillatory solutions.

In view of the above observation, we see that if Condition (2) fails to hold, then (E) cannot be reduced to a canonical-type equation; thus, the criteria established in [17,18] cannot be applicable to such types of equations. Therefore, the aim of this paper is to fill this gap. That is, we present another transform method that reduces (E) into a canonical form if ${\mathrm{\Omega }}_{21}\left(t_0\right)<\infty$. This significantly simplifies the examination for finding conditions for the oscillation of all solutions of Equation (E). Hence, the oscillation results obtained here are new and not only complement those in [11,17,18] but also contribute well to the oscillation theory of functional differential equations. Examples are provided to show the novelty and significance of our main results.

2. Main Results

For simplicity, we employ the following notation:

$${\mathrm{\Omega }}_1\left(t\right)={\int }_t^{\infty }{\displaystyle \frac{1}{{\delta }_1\left(s\right)}}ds,{\mathrm{\Omega }}_{21}\left(t\right)={\int }_t^{\infty }{\displaystyle \frac{{\mathrm{\Omega }}_1\left(s\right)}{{\delta }_2\left(s\right)}}ds,$$

$${\beta }_1\left(t\right)={\displaystyle \frac{{\delta }_1\left(t\right){\mathrm{\Omega }}_1^2\left(t\right)}{{\mathrm{\Omega }}_{21}\left(t\right)}},{\beta }_2\left(t\right)={\displaystyle \frac{{\delta }_2\left(t\right){\mathrm{\Omega }}_{21}^2\left(t\right)}{{\mathrm{\Omega }}_1\left(t\right)}},z\left(t\right)={\displaystyle \frac{\zeta \left(t\right)}{{\mathrm{\Omega }}_1\left(t\right)}}.$$

We begin with a theorem that is adopted from [14] but that is presented with a different proof.

Theorem 1.

If

$${\mathrm{\Omega }}_{21}\left(t_0\right)<\infty ,$$

(3)

then the semi-canonical operator $\mathfrak{L}\zeta$ has the following unique canonical representation:

$$\mathfrak{L}\zeta \left(t\right)={\displaystyle \frac{1}{{\mathrm{\Omega }}_{21}\left(t\right)}}{\left({\displaystyle \frac{{\delta }_2\left(t\right){\mathrm{\Omega }}_{21}^2\left(t\right)}{{\mathrm{\Omega }}_1\left(t\right)}}{\left({\displaystyle \frac{{\delta }_1\left(t\right){\mathrm{\Omega }}_1^2\left(t\right)}{{\mathrm{\Omega }}_{21}\left(t\right)}}{\left({\displaystyle \frac{\zeta \left(t\right)}{{\mathrm{\Omega }}_1\left(t\right)}}\right)}^{\prime }\right)}^{\prime }\right)}^{\prime }.$$

(4)

Proof. 

By a direct calculation, we have

$${\displaystyle \frac{{\delta }_2\left(t\right){\mathrm{\Omega }}_{21}^2\left(t\right)}{{\mathrm{\Omega }}_1\left(t\right)}}{\left({\displaystyle \frac{{\delta }_1\left(t\right){\mathrm{\Omega }}_1^2\left(t\right)}{{\mathrm{\Omega }}_{21}\left(t\right)}}{\left({\displaystyle \frac{\zeta \left(t\right)}{{\mathrm{\Omega }}_1\left(t\right)}}\right)}^{\prime }\right)}^{\prime }={\mathrm{\Omega }}_{21}\left(t\right){\delta }_2\left(t\right){\left({\delta }_1\left(t\right){\zeta }^{\prime }\left(t\right)\right)}^{\prime }+{\mathrm{\Omega }}_1\left(t\right){\delta }_1\left(t\right){\zeta }^{\prime }\left(t\right)+\zeta \left(t\right),$$

or

$$\begin{array}{c}{\displaystyle \frac{1}{{\mathrm{\Omega }}_{21}\left(t\right)}}{\left({\displaystyle \frac{{\delta }_2\left(t\right){\mathrm{\Omega }}_{21}^2\left(t\right)}{{\mathrm{\Omega }}_1\left(t\right)}}{\left({\displaystyle \frac{{\delta }_1\left(t\right){\mathrm{\Omega }}_1^2\left(t\right)}{{\mathrm{\Omega }}_{21}\left(t\right)}}{\left({\displaystyle \frac{\zeta \left(t\right)}{{\mathrm{\Omega }}_1\left(t\right)}}\right)}^{\prime }\right)}^{\prime }\right)}^{\prime }={\displaystyle \frac{1}{{\mathrm{\Omega }}_{21}\left(t\right)}}[{\mathrm{\Omega }}_{21}\left(t\right){\delta }_2\left(t\right){\left({\delta }_1\left(t\right){\zeta }^{\prime }\left(t\right)\right)}^{\prime }\hfill \\ \hfill +{\mathrm{\Omega }}_1\left(t\right){\delta }_1\left(t\right){\zeta }^{\prime }\left(t\right)+\zeta \left(t\right){]}^{\prime }\end{array}$$

$$\begin{array}{ccc}& =& {\displaystyle \frac{1}{{\mathrm{\Omega }}_{21}\left(t\right)}}[{\mathrm{\Omega }}_{21}\left(t\right){\left({\delta }_2\left(t\right){\left({\delta }_1\left(t\right){\zeta }^{\prime }\left(t\right)\right)}^{\prime }\right)}^{\prime }-{\mathrm{\Omega }}_1\left(t\right){\left({\delta }_1\left(t\right){\zeta }^{\prime }\left(t\right)\right)}^{\prime }\hfill \\ & +& {\mathrm{\Omega }}_1\left(t\right){\left({\delta }_1\left(t\right){\zeta }^{\prime }\left(t\right)\right)}^{\prime }-{\zeta }^{\prime }\left(t\right)+{\zeta }^{\prime }\left(t\right)]={\left({\delta }_2\left(t\right){\left({\delta }_1\left(t\right){\zeta }^{\prime }\left(t\right)\right)}^{\prime }\right)}^{\prime }.\hfill \end{array}$$

Next, we show that (4) is in a canonical form, that is,

$${\int }_{t_0}^{\infty }{\displaystyle \frac{{\mathrm{\Omega }}_1\left(t\right)}{{\delta }_2\left(t\right){\mathrm{\Omega }}_{21}^2\left(t\right)}}dt=\infty ={\int }_{t_0}^{\infty }{\displaystyle \frac{{\mathrm{\Omega }}_{21}\left(t\right)}{{\delta }_1\left(t\right){\mathrm{\Omega }}_1^2\left(t\right)}}dt.$$

Now,

$${\int }_{t_0}^{\infty }{\displaystyle \frac{{\mathrm{\Omega }}_1\left(t\right)}{{\delta }_2\left(t\right){\mathrm{\Omega }}_{21}^2\left(t\right)}}dt={\int }_{t_0}^{\infty }d\left({\displaystyle \frac{1}{{\mathrm{\Omega }}_{21}\left(t\right)}}\right)=\underset{t\to \infty }{lim}{\displaystyle \frac{1}{{\mathrm{\Omega }}_{21}\left(t\right)}}-{\displaystyle \frac{1}{{\mathrm{\Omega }}_{21}\left(t_0\right)}}=\infty .$$

Further,

$$\begin{array}{ccc}\hfill {\int }_{t_0}^{\infty }{\displaystyle \frac{{\mathrm{\Omega }}_{21}\left(t\right)}{{\delta }_1\left(t\right){\mathrm{\Omega }}_1^2\left(t\right)}}dt& =& {\int }_{t_0}^{\infty }{\mathrm{\Omega }}_{21}\left(t\right)d\left({\displaystyle \frac{1}{{\mathrm{\Omega }}_1\left(t\right)}}\right)\hfill \\ & =& {\displaystyle \frac{{\mathrm{\Omega }}_{21}\left(t\right)}{{\mathrm{\Omega }}_1\left(t\right)}}{|}_{t_0}^{\infty }+{\int }_{t_0}^{\infty }{\displaystyle \frac{1}{{\delta }_2\left(t\right)}}dt\hfill \\ & >& {\int }_{t_0}^{\infty }{\displaystyle \frac{1}{{\delta }_2\left(t\right)}}dt=\infty \hfill \end{array}$$

by $\left(H_5\right).$ However, in [19], Trench found that there exists only one canonical representation of $\mathfrak{L}$ (up to multiplicative constants with product 1); so, our canonical form is unique. The proof of the theorem is complete. □

Based on Theorem 1, one can write (E) in the canonical form as

$${\left({\beta }_2\left(t\right)({\beta }_1\left(t\right)z^{\prime }\left(t\right)\right)}^{\prime }{)}^{\prime }+w\left(t\right)z^{\alpha }\left(\tau \left(t\right)\right)=0,t\ge t_0,$$

(E_c)

where $w\left(t\right)=\eta \left(t\right){\mathrm{\Omega }}_{21}\left(t\right){\mathrm{\Omega }}_1^{\alpha }\left(\tau \left(t\right)\right),$ and the following results are immediate.

Theorem 2.

Let (3) hold. Then, the semi-canonical Equation (E) possesses a solution $\zeta \left(t\right)$ if and only if the canonical Equation (E) has the solution $z\left(t\right).$

Corollary 1.

Let (3) hold. The semi-canonical Equation (E) has an eventually positive solution if and only if the canonical Equation (E) has an eventually positive solution.

Corollary 1 significantly simplifies the investigation of (E) since, for (E_c), we deal with only two classes of an eventually positive (nonoscillatory) solution (see Lemma 2 [20]), namely, either

$$z\left(t\right)>0,{\mathcal{L}}_{1}z\left(t\right)<0,{\mathcal{L}}_{2}z\left(t\right)>0,{\mathcal{L}}_{3}z\left(t\right)<0,$$

and, in this case, we say $z\in D_0$ or

$$z\left(t\right)>0,{\mathcal{L}}_{1}z\left(t\right)>0,{\mathcal{L}}_{2}z\left(t\right)>0,{\mathcal{L}}_{3}z\left(t\right)<0,$$

and, in this case, we denote that $z\in D_2,$ where

$${\mathcal{L}}_{0}z\left(t\right)=z\left(t\right),{\mathcal{L}}_{i}z\left(t\right)={\beta }_i\left(t\right){\left({\mathcal{L}}_{i-1}z\left(t\right)\right)}^{\prime },i=1,2,{\mathcal{L}}_{3}z\left(t\right)={\left({\mathcal{L}}_{2}z\left(t\right)\right)}^{\prime }.$$

Theorem 3.

Let (3) hold. Assume that there exists a function $\sigma \in C^1\left([t_0,\infty )\right)$ such that

$${\sigma }^{\prime }\left(t\right)>0,\sigma \left(t\right)>t,g\left(t\right)=\tau \left(\sigma \left(\sigma \left(t\right)\right)\right)<t.$$

(5)

If both the first-order delay differential equations

$$u^{\prime }\left(t\right)+Q_1\left(t\right)u^{\alpha }\left(\tau \left(t\right)\right)=0$$

(6)

and

$$u^{\prime }\left(t\right)+Q_2\left(t\right)u^{\alpha }\left(g\left(t\right)\right)=0,$$

(7)

where

$$\begin{array}{ccc}\hfill Q_1\left(t\right)& =& w\left(t\right){\left({\int }_{t_1}^{\tau \left(t\right)}{\displaystyle \frac{1}{{\beta }_1\left(s\right)}}{\int }_{t_1}^s{\displaystyle \frac{1}{{\beta }_2\left(s_1\right)}}d{s}_{1}ds\right)}^{\alpha }\hfill \\ \hfill Q_2\left(t\right)& =& {\displaystyle \frac{1}{{\beta }_1\left(t\right)}}{\int }_t^{\sigma \left(t\right)}{\displaystyle \frac{1}{{\beta }_2\left(s\right)}}{\int }_s^{\sigma \left(s\right)}w\left(s_1\right)d{s}_{1}ds\hfill \end{array}$$

for all $t_1\ge t_0,$ are oscillatory, then Equation (E) is oscillatory.

Proof. 

Let $\zeta \left(t\right)$ be an eventually positive solution of (E). Then, by Corollary 1, $z\left(t\right)$ is also a positive solution of (E_c) and either $z\left(t\right)\in D_0$ or $z\left(t\right)\in D_2$ for all $t\ge t_1\ge t_0.$

First, assume that $z\left(t\right)\in D_2.$ Then, using the fact that ${\mathcal{L}}_{2}z\left(t\right)>0$ and decreasing, we have

$${\mathcal{L}}_{1}z\left(t\right)\ge {\int }_{t_1}^t{\displaystyle \frac{{\mathcal{L}}_{2}z\left(s\right)}{{\beta }_2\left(s\right)}}ds\ge {\mathcal{L}}_{2}z\left(t\right){\int }_{t_1}^t{\displaystyle \frac{1}{{\beta }_2\left(s\right)}}ds,$$

or

$$z^{\prime }\left(t\right)\ge {\mathcal{L}}_{2}z\left(t\right)\left({\displaystyle \frac{1}{{\beta }_1\left(t\right)}}{\int }_{t_1}^t{\displaystyle \frac{1}{{\beta }_2\left(s\right)}}ds\right).$$

Integrating again from $t_1$ to $t,$ we get

$$z\left(t\right)\ge {\mathcal{L}}_{2}z\left(t\right){\int }_{t_1}^t{\displaystyle \frac{1}{{\beta }_1\left(s\right)}}\left({\int }_{t_1}^s{\displaystyle \frac{1}{{\beta }_2\left(s_1\right)}}d{s}_1\right)ds.$$

(8)

Let $u\left(t\right)={\mathcal{L}}_{2}z\left(t\right).$ Then, combining (8) with (E_c), we see that

$$u^{\prime }\left(t\right)+w\left(t\right){\left({\int }_{t_1}^{\tau \left(t\right)}{\displaystyle \frac{1}{{\beta }_1\left(s\right)}}{\int }_{t_1}^s{\displaystyle \frac{1}{{\beta }_2\left(s_1\right)}}d{s}_{1}ds\right)}^{\alpha }{u}^{\alpha }\left(\tau \left(t\right)\right)\le 0$$

for $t\ge t_1.$ By integrating the latter inequality from t to $\infty ,$ we have

$$u\left(t\right)\ge {\int }_t^{\infty }{Q}_1\left(s\right)u^{\alpha }\left(\tau \left(s\right)\right)ds$$

for $t\ge t_1.$ The function $u\left(t\right)$ is clearly decreasing on $[t_1,\infty )$; hence, with Theorem 1 in [21], we conclude that there exists a positive solution $u\left(t\right)$ of (6) with ${lim}_{t\to \infty }u\left(t\right)=0,$ which contradicts the fact that (6) is oscillatory.

Next, assume that $z\left(t\right)\in D_0.$ Integrating (E_c) from t to $\sigma \left(t\right)$ gives

$${\mathcal{L}}_{2}z\left(t\right)\ge {\int }_t^{\sigma \left(t\right)}w\left(s\right)z^{\alpha }\left(\tau \left(s\right)\right)ds\ge z^{\alpha }\left(\tau \left(\sigma \left(t\right)\right)\right){\int }_t^{\sigma \left(t\right)}w\left(s\right)ds.$$

Then,

$${\left({\beta }_1\left(t\right)z^{\prime }\left(t\right)\right)}^{\prime }\ge {\displaystyle \frac{z^{\alpha }\left(\tau \left(\sigma \left(t\right)\right)\right)}{{\beta }_2\left(t\right)}}{\int }_t^{\sigma \left(t\right)}w\left(s\right)ds.$$

By integrating the last inequality from t to $\sigma \left(t\right),$ we get

$$-z^{\prime }\left(t\right)\ge {\displaystyle \frac{z^{\alpha }\left(g\left(t\right)\right)}{{\beta }_1\left(t\right)}}{\int }_t^{\sigma \left(t\right)}{\displaystyle \frac{1}{{\beta }_2\left(s\right)}}{\int }_s^{\sigma \left(s\right)}w\left(s_1\right)d{s}_{1}ds.$$

Finally, integrating from t to ∞ yields

$$z\left(t\right)\ge {\int }_t^{\infty }{\displaystyle \frac{z^{\alpha }\left(g\left(s\right)\right)}{{\beta }_1\left(s\right)}}{\int }_s^{\sigma \left(s\right)}{\displaystyle \frac{1}{{\beta }_2\left(s_1\right)}}{\int }_{s_1}^{\sigma \left(s_1\right)}w\left(s_2\right)d{s}_{2}d{s}_{1}ds.$$

Set the right-hand side of the last inequality with $u\left(t\right)$; thus, we have $z\left(t\right)\ge u\left(t\right)>0.$ Therefore, it is easy to see that

$$0=u^{\prime }\left(t\right)+Q_2\left(t\right)z^{\alpha }\left(g\left(t\right)\right)$$

$$0\ge u^{\prime }\left(t\right)+Q_2\left(t\right)u^{\alpha }\left(g\left(t\right)\right).$$

Since $u\left(t\right)$ is a positive bounded solution of the last inequality, then, by Corollary 1 of [21], we see that the corresponding differential Equation (7) has also a positive solution. This is a contradiction to our assumption, and we conclude that (E) oscillates. The proof of the theorem is complete. □

By employing criteria for the oscillation of (6) and (7), we immediately obtain explicit criteria for the oscillation of (E) for a different value of $\alpha .$

Corollary 2.

Let (3) hold. Assume that there exists a function $\sigma \in C^1\left([t_0,\infty )\right)$ such that (5) holds. If $\alpha =1,$

$$\underset{t\to \infty }{lim}inf{\int }_{\tau \left(t\right)}^t{Q}_1\left(s\right)ds>{\displaystyle \frac{1}{e}}$$

(9)

and

$$\underset{t\to \infty }{lim}inf{\int }_{g\left(t\right)}^t{Q}_2\left(s\right)ds>{\displaystyle \frac{1}{e}},$$

(10)

then (E) is oscillatory.

Proof. 

The application of Theorem 2.1.1 of [2] with (9) and (10) implies that Equations (6) and (7) are oscillatory. Now, the proof follows from Theorem 3. This ends the proof. □

Corollary 3.

Let (3) hold. Assume that there exists a function $\sigma \in C^1\left([t_0,\infty )\right)$ such that (5) holds. If $0<\alpha <1,$

$${\int }_{t_{\ast }}^{\infty }{Q}_1\left(t\right)dt=\infty$$

(11)

and

$${\int }_{t_{\ast }}^{\infty }{Q}_2\left(t\right)dt=\infty$$

(12)

for all $t_{\ast }\ge t_0,$ then Equation (E) is oscillatory.

Proof. 

The application of Theorem 3.9.3 of [2] with (11) and (12) implies that Equations (6) and (7) are oscillatory. Now, the proof follows from Theorem 3. This ends the proof. □

Corollary 4.

Let (3) hold. Assume that there exists a function $\sigma \in C^1\left([t_0,\infty )\right)$ such that (5) holds. Further, assume that $\alpha >1,$ $\tau \left(t\right)={\theta }_{1}t,$ $\sigma \left(t\right)={\theta }_2\left(t\right),$ ${\theta }_1\in (0,1),$ ${\theta }_2>1$ and $g\left(t\right)={\theta }_3\left(t\right)$ with ${\theta }_3={\theta }_1{\theta }_2^2<1.$ If there exist

$${\lambda }_1>-ln\alpha /ln{\theta }_1,{\lambda }_2>-ln\alpha /ln{\theta }_3$$

(13)

such that

$$\underset{t\to \infty }{lim}inf[Q_1\left(t\right)exp(-t^{{\lambda }_1})]>0$$

(14)

and

$$\underset{t\to \infty }{lim}inf[Q_2\left(t\right)exp(-t^{{\lambda }_2})]>0$$

(15)

hold, then (E) is oscillatory.

Corollary 5.

Let (3) hold. Assume that there exists a function $\sigma \in C^1\left([t_0,\infty )\right)$ such that (5) holds. Further, assume that $\alpha >1,$ $\tau \left(t\right)=t^{{\theta }_1},$ $\sigma \left(t\right)=t^{{\theta }_2},$ ${\theta }_1\in (0,1),$ ${\theta }_2>1$ and $g\left(t\right)=t^{{\theta }_3}$ with ${\theta }_3={\theta }_1{\theta }_2^2<1.$ If there exist

$${\lambda }_1>-ln\alpha /ln{\theta }_1,{\lambda }_2>-ln\alpha /ln{\theta }_3$$

(16)

such that

$$\underset{t\to \infty }{lim}inf[Q_1\left(t\right)exp(-{(lnt)}^{{\lambda }_1})]>0$$

(17)

and

$$\underset{t\to \infty }{lim}inf[Q_2\left(t\right)exp(-{(lnt)}^{{\lambda }_2})]>0$$

(18)

hold, then (E) is oscillatory.

The proofs of Corollary 4 and Corollary 5 follow by using Theorems 4 and 5 of [22] with Theorem 3, respectively.

Next, we present another criteria for the oscillation of (E) when $\alpha =1$ and $\alpha <1.$

Theorem 4.

Let (3) and $\alpha =1$ hold. If

$$\underset{t\to \infty }{lim}sup{\int }_{\tau \left(t\right)}^t\left({\displaystyle \frac{1}{{\beta }_1\left(s\right)}}{\int }_s^t{\displaystyle \frac{1}{{\beta }_2\left(s_1\right)}}{\int }_{s_1}^{t}w\left(s_2\right)d{s}_{2}d{s}_1\right)ds>1$$

(19)

and

$$\underset{t\to \infty }{lim}inf{\int }_{\tau \left(t\right)}^t{Q}_1\left(s\right)ds>{\displaystyle \frac{1}{e}},$$

(20)

then Equation (E) is oscillatory.

Proof. 

Let $\zeta \left(t\right)$ be an eventually positive solution of (E). Then, proceeding as in the proof of Theorem 2.4, we see that $z\left(t\right)$ is a positive solution of (E_c) and belongs to either $D_0$ or $D_2$ for all $t\ge t_1\ge t_0.$

If $z\left(t\right)\in D_2,$ then as in the Proof of Corollary 2, we obtain a contradiction with (20); thus, $z\left(t\right)\in D_0.$

Integrating (E_c) from s to t gives

$$\begin{array}{ccc}\hfill {\left({\beta }_1\left(s\right)z^{\prime }\left(s\right)\right)}^{\prime }& \ge & {\displaystyle \frac{1}{{\beta }_2\left(s\right)}}{\int }_s^{t}w\left(s_1\right)z^{\alpha }\left(\tau \left(s_1\right)\right)d{s}_1\hfill \\ & \ge & {\displaystyle \frac{z^{\alpha }\left(\tau \left(t\right)\right)}{{\beta }_2\left(s\right)}}{\int }_s^{t}w\left(s_1\right)d{s}_1.\hfill \end{array}$$

Again, by integrating the inequality twice from s to $t,$ we have

$$z\left(s\right)\ge z^{\alpha }\left(\tau \left(t\right)\right){\int }_s^t\left({\displaystyle \frac{1}{{\beta }_1\left(s_1\right)}}{\int }_{s_1}^t{\displaystyle \frac{1}{{\beta }_2\left(s_2\right)}}{\int }_{s_2}^{t}w\left(s_3\right)d{s}_{3}d{s}_2\right)d{s}_1.$$

By letting $s=\tau \left(t\right)$ and $\alpha =1,$ we obtain a contradiction with (19). Hence, we conclude that (E) is oscillatory. This completes the proof. □

Theorem 5.

Let (3) and $0<\alpha <1$ hold. If

$$\underset{t\to \infty }{lim}sup{\int }_{\tau \left(t\right)}^t\left({\displaystyle \frac{1}{{\beta }_1\left(s\right)}}{\int }_s^t{\displaystyle \frac{1}{{\beta }_2\left(s_1\right)}}{\int }_{s_1}^{t}w\left(s_2\right)d{s}_{2}d{s}_1\right)ds=\infty$$

(21)

and

$$\underset{t\to \infty }{lim}{\int }_{t_1}^{\infty }{Q}_1\left(s\right)ds=\infty$$

(22)

for all $t_1\ge t_0,$ then Equation (E) is oscillatory.

Proof. 

Let $\zeta \left(t\right)$ be an eventually positive solution of (E). Then, by proceeding as in the proof of Theorem 3, we see that $z\left(t\right)$ is a positive solution of (E_c) that belongs to either $D_0$ or $D_2$ for all $t\ge t_1\ge t_0.$

First, assume that $z\left(t\right)\in D_2.$ In view of Condition (22) and by Corollary 3, we understand that $D_2$ is empty; therefore, $z\left(t\right)\in D_0.$ Thus, by proceeding as in the proof of Theorem 4, we obtain

$$z^{1-\alpha }\left(\tau \left(t\right)\right)\ge {\int }_{\tau \left(t\right)}^t\left({\displaystyle \frac{1}{{\beta }_1\left(s\right)}}{\int }_s^t{\displaystyle \frac{1}{{\beta }_2\left(s_1\right)}}{\int }_{s_1}^{t}w\left(s_2\right)d{s}_{2}d{s}_1\right)ds.$$

(23)

Since $z\left(t\right)$ is decreasing and $\alpha <1,$ we see that $z^{1-\alpha }\left(\tau \left(t\right)\right)\le M$ for all $t\ge t_1\ge t_0$. Using this in (23), one obtains a contradiction with (21). The proof of the theorem is complete. □

Remark 1.

Here, we use the canonical Equation (E_c) to obtain oscillation criteria of (E). Therefore, it is clear that one may use the results in [7,8,9,12,13,15,16] to obtain several oscillatory and asymptotic behaviors of (E_c), which in turn imply that of (E). The details are left to the reader.

3. Examples

In this section, we present three examples to show the importance over the already known results in [17,18] in the sense that the condition ${\mathrm{\Omega }}_{21}\left(t_0\right)=\infty$ fails to hold.

Example 1.

Consider the following semi-canonical third-order linear delay differential equation:

$${\left(t{\left(t^2{\zeta }^{\prime }\left(t\right)\right)}^{\prime }\right)}^{\prime }+{\eta }_0\zeta \left(\lambda t\right)=0,t\ge 1,$$

(24)

where ${\eta }_0>0$ and $\lambda \in (0,1).$

Here, ${\delta }_1\left(t\right)=t^2,$ ${\delta }_2\left(t\right)=t,$ $\eta \left(t\right)={\eta }_0,$ $\tau \left(t\right)=\lambda t$ and $\alpha =1.$ Via a simple calculation, we have ${\mathrm{\Omega }}_1\left(t\right)={\displaystyle \frac{1}{t}},$ ${\mathrm{\Omega }}_{21}\left(t\right)={\displaystyle \frac{1}{t}},$ ${\beta }_1\left(t\right)=t,$ ${\beta }_2\left(t\right)=1$ and $w\left(t\right)={\displaystyle \frac{{\eta }_0}{\lambda t^2}}.$ So, Condition (3) holds, and the transformed equation is

$${\left(t{z}^{\prime }\left(t\right)\right)}^{″}+{\displaystyle \frac{{\eta }_0}{{\lambda }^{2}t}}z\left(\lambda t\right)=0,t\ge 1,$$

which is in a canonical form. Choose $\sigma \left(t\right)=\mu t$ with $1<\mu <{\displaystyle \frac{1}{\sqrt{\lambda }}}$ and $g\left(t\right)=\lambda {\mu }^{2}t<t$; thus, we see that

$$Q_1\left(t\right)={\eta }_0\left({\displaystyle \frac{1}{t}}-{\displaystyle \frac{1}{\lambda t^2}}-{\displaystyle \frac{ln\lambda }{\lambda t^2}}-{\displaystyle \frac{lnt}{\lambda t^2}}\right),$$

$$Q_2\left(t\right)={\displaystyle \frac{{\eta }_0}{\lambda t}}\left(1-{\displaystyle \frac{\lambda }{\mu }}\right)ln\mu .$$

Condition (9) becomes

$${\eta }_{0}ln{\displaystyle \frac{1}{\lambda }}>{\displaystyle \frac{1}{e}}.$$

Condition (10) becomes

$${\displaystyle \frac{{\eta }_0}{\lambda }}\left(1-{\displaystyle \frac{1}{\mu }}\right)ln\mu ln{\displaystyle \frac{1}{\lambda {\mu }^2}}>{\displaystyle \frac{1}{e}}.$$

Therefore, by Corollary 2, Equation (24) is oscillatory if

$${\eta }_0>max\left\{{\displaystyle \frac{1}{eln\frac{1}{\lambda }}},{\displaystyle \frac{\lambda \mu }{(\mu -1)ln\mu ln\frac{1}{\lambda {\mu }^2}}}\right\}.$$

Example 2.

Consider the following third-order semi-canonical sub-linear delay differential equation:

$${\left(t^3{\zeta }^{\prime }\left(t\right)\right)}^{″}+{\eta }_{0}t{\zeta }^{{\displaystyle \frac{1}{3}}}\left(\lambda t\right)=0,t\ge 1,$$

(25)

where ${\eta }_0>1$ and $\lambda \in (0,1).$

Here, ${\delta }_2\left(t\right)=1,$ ${\delta }_1\left(t\right)=t^3,$ $\eta \left(t\right)={\eta }_{0}t,$ $\tau \left(t\right)=\lambda t$ and $\alpha ={\displaystyle \frac{1}{3}}.$ Via a simple calculation, we have ${\mathrm{\Omega }}_1\left(t\right)={\displaystyle \frac{1}{2t^2}},$ ${\mathrm{\Omega }}_{21}\left(t\right)={\displaystyle \frac{1}{2t}},$ ${\beta }_1\left(t\right)={\beta }_2\left(t\right)={\displaystyle \frac{1}{2}},$ $w\left(t\right)={\displaystyle \frac{{\eta }_1}{t^{\frac{2}{3}}}}$ where ${\eta }_1={\eta }_0{\left({\displaystyle \frac{2}{\lambda }}\right)}^{{\displaystyle \frac{2}{3}}}.$ Condition (3) clearly holds, and the transformed equation is

$$z^{‴}\left(t\right)+{\displaystyle \frac{\eta }{t^{\frac{2}{3}}}}{z}^{{\displaystyle \frac{1}{3}}}\left(\lambda t\right)=0,t\ge 1,$$

which is in a canonical form. With a further calculation, we see that

$$Q_1\left(t\right)=2^{{\displaystyle \frac{1}{3}}}{\eta }_0.$$

Choose $\sigma \left(t\right)=\mu t$ with $1<\mu <{\displaystyle \frac{1}{\lambda }}$; thus, we see that $g\left(t\right)=\lambda {\mu }^{2}t<t$. Therefore, Condition (5) holds. Also,

$$Q_2\left(t\right)={\displaystyle \frac{9}{4}}{\eta }_1{(\mu -1)}^{{\displaystyle \frac{5}{3}}}{t}^{{\displaystyle \frac{4}{3}}}.$$

Conditions (11) and (12) clearly hold. Therefore, by Corollary 3, Equation (25) is oscillatory for all ${\eta }_0>0.$

Example 3.

Consider the following third-order semi-canonical super-linear delay differential equation:

$${\left(t^4{\zeta }^{\prime }\left(t\right)\right)}^{″}+{\eta }_0{t}^{12}{e}^{t^2}{\zeta }^3\left({\displaystyle \frac{t}{4}}\right)=0,t\ge 1,$$

(26)

where ${\eta }_0>0.$

Here, ${\delta }_1\left(t\right)=t^4,$ ${\delta }_2\left(t\right)=1,$ $\eta \left(t\right)={\eta }_0{t}^{12},$ $\tau \left(t\right)={\displaystyle \frac{t}{4}}$ and $\alpha =3.$ Via a simple computation, we see that

$${\mathrm{\Omega }}_1\left(t\right)={\displaystyle \frac{1}{3t^3}}and{\mathrm{\Omega }}_{21}\left(t\right)={\displaystyle \frac{1}{6t^2}}.$$

Thus, Condition (3) is satisfied. The transformed equation is

$${\left({\displaystyle \frac{1}{t}}{z}^{″}\left(t\right)\right)}^{\prime }+{\eta }_{1}t{e}^{t^2}{z}^3\left({\displaystyle \frac{t}{4}}\right)=0,t\ge 1,$$

where ${\eta }_1={\displaystyle \frac{{\eta }_0{\left(4\right)}^9}{9}}.$ Choose $\sigma \left(t\right)={\displaystyle \frac{3}{2}}t$; thus, we have $g\left(t\right)={\displaystyle \frac{9}{16}}t$. Therefore, Condition (5) holds. With a further calculation, we see that

$$Q_1\left(t\right)={\displaystyle \frac{{\eta }_0}{9\left(6^3\right)}}{t}^{10}{e}^{t^2},Q_2\left(t\right)={\displaystyle \frac{{\eta }_1}{36}}\left(4e^{{\displaystyle \frac{81}{16}}{t}^2}-13e^{{\displaystyle \frac{9}{4}}{t}^2}+9e^{t^2}\right).$$

Let ${\lambda }_1=1$; thus, $1>{\displaystyle \frac{ln3}{ln4}}$. Moreover, let ${\lambda }_2=2$; thus, $2>{\displaystyle \frac{ln3}{ln\left(\frac{16}{9}\right)}}.$ Therefore, the conditions in (13) hold. Condition (14) becomes

$$\underset{t\to \infty }{lim}inf\left[{\displaystyle \frac{{\eta }_0}{9\left(6^3\right)}}{t}^{10}\left(e^{t^2-t}\right)\right]>0,$$

that is, Condition (14) holds. Condition (15) becomes

$$\underset{t\to \infty }{lim}inf\left[{\displaystyle \frac{{\eta }_1}{36}}\left(4e^{{\displaystyle \frac{81}{16}}{t}^2}-13e^{{\displaystyle \frac{9}{4}}{t}^2}+9e^{t^2}\right)e^{-t^2}\right]>\underset{t\to \infty }{lim}{\displaystyle \frac{{\eta }_1}{36}}\left(3e^{{\displaystyle \frac{5}{4}}{t}^2}+9-16e^{-t^2}\right)>0,$$

that is, Condition (15) holds. Therefore, by Corollary 4, Equation (26) is oscillatory.

4. Conclusions

In this paper, by using a new canonical transform, we changed the shape of Equation (E) into a canonical-type equation. This technique reduced the number of classes of nonoscillatory solutions into two instead of three. By applying a comparison with first-order delay differential equations, we were able to eliminate these two classes of nonoscillatory solutions to obtain the criteria for the oscillation of all solutions of Equation (E). Also, the importance and the significance of the results obtained here were verified through three examples.

Note that the results in [9,10,12,13,14,15,16,17,18] cannot be applied to the above-mentioned examples to obtain any conclusions since the equations in these examples are semi-canonical and of a different type. Therefore, the results established here are novel and complement those mentioned in the references.

Furthermore, in light of future research directions, we present the potential for extending the scope of this study. Specifically, we propose exploring the application of this method to establish criteria for determining the oscillatory or asymptotic behavior of solutions of third-order semi-canonical neutral-type functional differential equations and higher-order functional differential equations. Such an extension of our study could significantly enhance the applicability of the techniques used in this paper to a wide range of differential equations.